Ch 10.3: Polar Coordinates

In this section, we will

1. define the polar coordinate system

2. convert points in polar coordinates to Cartesian

3. convert points in Cartesian to polar

4. graph simple polar curves

Polar coordinate system

I The origin is also called the pole

I The polar axis corresponds to the x-axis

Polar coordinate system (when r < 0)

I (r , θ) and (−r , θ) lie on the same line through O and at thesame distance from O, but on opposite side of O

ExamplesPlot the points whose polar coordinates are as follows:(a) (1, 3π) (b) (1, 5π/4) (c) (2,−5π/4) (d) (−3, 3π/4)

Multiple representations of a pointHow many ways can we represent (1, 5π/4) in polar coordinates?

Conversion between polar and Cartesian coordinates

Conversion between polar and Cartesian coordinates

I From polar to Cartesian: x = r cos θ, y = r sin θ

I From Cartesian to polar: r2 = x2 + y2, tan θ = yx

Examples)

1. Convert (2, π/3) from polar to Cartesian coordinates.

2. Convert (1,−1) from Cartesian to polar coordinates.

Polar Curves

What curve is represented by r = 1? How about r = 2?

Polar Curves

What curve is represented by θ = 1?

Polar Curves1. Sketch the curve with polar equation r = cos θ. How about

r = 2 cos θ?2. Find a Cartesian equation for this curve.

Polar Curves

What curve is represented by r = 2 sin θ?

Polar Curves : CardioidWhat curve is represented by r = 1 + sin θ?

r = 1 + sin θ : Cardioid

Polar Curves : four-leaved rose

Sketch the curve r = cos 2θ.